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Graph Theoretical and Algorithmic Characterizations of Positive Definite Symmetric Quasi-Cartan Matrices

Abarca M., Rivera D.

Fundamenta Informaticae

2016

Vol. 149, nr 3

241--261

A well known constructive proof for the ADE-classification of many mathematical objects, such as positive unit forms and their associated quasi-Cartan matrices, has lead to an Inflations Algorithm. However, this algorithm is not known to run in polynomial time. In this paper we use a so called flation transformation and show how its invariants can be used to characterize the Dynkin types A and D in the language of graph theory. Also, a polynomial-time algorithm for computing the Dynkin type is suggested.